Estimates for the Derivatives of the Poisson Kernel on Nilpotent Meta-Abelian Groups

Abstract

Let S be a semi direct product \(S=N\rtimes A\) where N is a connected and simply connected, non-abelian, nilpotent meta-abelian Lie group and A is isomorphic with ℝk, k > 1. We consider a class of second order left-invariant differential operators on S of the form \(\mathcal{L}_\alpha=L^a+\Delta_\alpha,\) where α ∈ ℝk, and for each a ∈ ℝk, La is left-invariant second order differential operator on N and \(\Delta_\alpha=\Delta-\langle\alpha,\nabla\rangle,\) where Δ is the usual Laplacian on ℝk. Using some probabilistic techniques (skew-product formulas for diffusions on S and N respectively, the concept of the derivative of a measure, etc.) we obtain an upper bound for the derivatives of the Poisson kernel for \(\mathcal{L}_\alpha.\) During the course of the proof we also get an upper estimate for the derivatives of the transition probabilities of the evolution on N generated by Lσ(t), where σ is a continuous function from [0, ∞ ) to ℝk.